Factor the following trinomials completely.
$z^2-11z+30$

• A. $\(\left\)(z+5\(\right\))\(\left\)(z-6\(\right\))$
• B. $\(\left\)(z-3\(\right\))\(\left\)(z-10\(\right\))$
• C. $\(\left\)(z+5\(\right\))\(\left\)(z+6\(\right\))$
• D. $\(\left\)(z-5\(\right\))\(\left\)(z-6\(\right\))$

Answer: (D)

Verified step by step guidance

1. Identify the trinomial to factor: \(z^2 - 11z + 30\).
2. Look for two numbers that multiply to the constant term \(30\) and add up to the coefficient of the middle term, which is \(-11\).
3. List the factor pairs of \(30\): \((1, 30)\), \((2, 15)\), \((3, 10)\), and \((5, 6)\), and consider their signs to sum to \(-11\).
4. Since the middle term is negative and the constant is positive, both numbers must be negative. Check which pair of negative numbers multiply to \(30\) and add to \(-11\): \(-5\) and \(-6\).
5. Write the factored form using these numbers: \((z - 5)(z - 6)\).